Characterizing the Fourier Transform by Phase and Amplitude Preserving Maps

It is shown that under the Fourier transform $F:L^2\left(R\right)\to L^2\left(R\right)$ of an even or odd function $\phi :R\to C$, the invariance of $|\phi |$ leads to the invariance or minus invariance of $\arg \phi$, and that the inverse holds in an analogous way. Such a relation between the phase and amplitude does not necessarily hold for other integral transforms. This implies that the Fourier transform can be characterized by the phase and amplitude preserving maps.